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yiACHIKERTS REFERENCE 3E. -MO. 53 PUEMoHE. ", : T M/ CHINERY, NEW

MACHINERY'S REFERENCE SERIES

EACH NUMBER IS A UNIT IN A SERIES ON ELECTRICAL AND

STEAM ENGINEERING DRAWING AND MACHINE

DESIGN AND SHOP PRACTICE

NUMBER 53

USE OF LOGARITHMS AND LOGARITHMIC TABLES

SECOND EDITION

CONTENTS

The Use of Logarithms, by ERIK OBERG - - 3

Tables of Logarithms - - 18

Copyright, 1912, The Industrial Press, Publishers of MACHINERY, 49-55 Lafayette Street, New York City

THE USB OP LOGARITHMS

It is not intended in the following pages to discuss the mathematical principles on which logarithms and expressions containing logarithms are based, but simply to impart a working knowledge of the use of logarithms, so that practical men, unfamiliar with this means for eliminating much of the work ordinarily required in long and cumber- some calculations, may be able to make advantageous use of the tables of logarithms given in the latter part of the book.

The object of logarithms is to facilitate and shorten calculations involving multiplication, division, x the extraction of roots, and the obtaining of powers of numbers, as will be explained later; but ordin- ary logarithms cannot be applied to operations involving addition and subtraction. Before entering directly upon the subject of the use of logarithms in carrying out the various classes of calculations men- tioned, it will be necessary to deal with the question of how to find the logarithm of a given number from the tables; or, if the logarithm is given, how to find the corresponding number.

A logarithm consists of two parts, a whole number and a decimal. The whole number, which may be either a positive or a negative num- ber,* or zero, according to the rules which will be given in the follow- ing, is called the characteristic; the decimal is called the mantissa. The decimal or mantissa only is given in the tables of logarithms on pages 18 to 35, inclusive, and is always positive. The logarithm of 350, for example is 2.54407. Here "2" is the characteristic, and "54407" is the mantissa, this latter being found from the table on page 23.

Rules for Finding- the Characteristic

The characteristic is not given in the tables of logarithms, due partly to the fact that it can so easily be determined without the aid of tables, and partly because the tables, so to say, are universal when the characteristic is left out.

For 1 and all numbers greater than 1 the characteristic is one less than the number of places to the left of the decimal point in the given number.

The characteristic of the logarithm of 237, therefore, is 2, because 2 is one less than the number of figures in 237. The characteristic of the logarithm of 237.26 is also 2, because it is only the number of figures to the left of the decimal point that is considered.

The characteristic of the logarithm of 7 is 0, because 0 is one less

* See MACHINERY'S Reference Series No. 54, Solution of Triangles, Chapter III, Positive and Negative Quantities.

347608

4 No.. 53—UVE O'F LOGARITHMS

than 1, which is the number of places in the given number. Below are given several numbers with the characteristics of their logarithms:

Characteristic of Number Logarithm

31 1

3163 3

229.634 2

1,112,352.62 6

1000 3

100 2

1 0

For numbers smaller than 1, that is, for numbers wholly decimal, the characteristic is negative, and its numerical value is one more tJian the number of ciphers between the decimal point and the first decimal which is not a cipher.

The numerical value of the characteristic of the logarithm of 0.036, therefore, is 2; being negative it is — 2. Instead of writing the minus sign ( — ) in front of or before the figure ( — 2), it is, however, written over the figure ( 2 ) • This method is used because the minus sign refers only to the characteristic, and not to the mantissa, this latter always being positive. In the same way, the characteristic of the logarithm of 0.36 is I ; and the characteristic of the logarithm of 0.0006 is i. Below are given several examples :

Characteristic of Number Logarithm

0.00.00275 5

0.3 I

0.375 T

0.000812 4

0.01234 5

Finding1 the Logarithms of Numbers

After the characteristic has been found by the rules just given, the mantissa must be found from the tables of logarithms. When finding the mantissa the decimal point is entirely disregarded. The mantissa of the logarithms of 2716, 271.6, 27.16, 2.716, or 0.02716, for example, is the same; it is only necessary to find the given figures in the tables, irrespective of the location of the decimal point.

Referring now to the tables on pages 18 to 35, it will be seen that numbers from 100 to 1,000 are given in the left-hand column. In addition, at the top of the tables, are figures from 0 to 9, each heading a column of logarithms. These additional figures make it possible to obtain directly from the tables the logarithms for all numbers with four figures or less. The body of the tables gives the mantissa of the logarithms.

While the tables do not give directly the mantissa of logarithms of numbers with more than four figures, it is possible to approximate the logarithm for numbers with a greater number of figures by methods which will be explained later. At present, when the logarithm is required for numbers with five or more figures, we will assume that

METHODS, RULES AND EXAMPLES 5

for practical results it is accurate enough to find the mantissa of the logarithm of the first four figures of the number, remembering, of course, that if the fifth figure is more than 5, then the fourth figure should be increased by one unit.

To find the logarithm of a number from the tables, first locate the three first figures of the number for which the logarithm is required in the left-hand column, and then find the fourth figure at the top of the columns of the page. Then follow the column down from this last figure until opposite the three first figures in the left-hand column. The figure thus found in the body of the table is the mantissa of the logarithm, the characteristic having already been found by the rules previously given.

If the number of which the logarithm is required does not contain four figures, annex ciphers to the right so as to obtain four figures. If the mantissa of the logarithm of 6 is required, for example, find the mantissa for 6,000. The mantissa is the same for 6, 0.6, 0.06, 60, 600, 6,000, etc., as already explained. The difference in the logarithm is taken care of by the change of characteristic for these various values. In order to save space in the tables, it will be seen by referring to them that the first two figures of the mantissa have been given in the first columns of logarithms only, the 0-column. These two figures should, however, always precede the three figures given in each of the following columns.

A few examples will now make the use of the tables clearer.

Example 1. — Find the logarithm of 1852.

Following the rule already given, locate 185 in the left-hand column of the tables (it will be found on page 19), and then following down- ward the column headed "2" at the top of the page, find the required mantissa opposite 185. It will be seen that the mantissa is .26764,* the figures 26 being found in the column under "0" and prefixed to the figures 764 found directly in the column under "2." The charac- teristic of the logarithm, according to the rules previously given, is 3. Hence the logarithm of 1852, or, as it is commonly written, log 1852 — 3.26764.

Example 2. — Find log 1.852.

As the figures in this number are the same as in that given in ex- ample 1, the mantissa remains the same; but the characteristic is 0. Therefore, the required logarithm, or log 1.852 = 0.26764.

Example 3. — Find log 93.14.

Locate 931 in the left-hand column of the tables (page 34), and then following downward the column headed "4" at the top of the page, find the required mantissa opposite 931. It will be found that the mantissa is .96914. The characteristic is 1. Hence log 93.14 = 1.96914.

Example 4. — Find log 4.576.

Find as before the last three figures of the mantissa opposite 457

* All the mantissas, or the numbers in the tables, are decimals, and the decimal point has, therefore, been omitted entirely, since no confusion could arise from this; but it should always be put before the figures of the mantissa as soon as taken from the table. The practice of eliminating the decimal point from the tables is common to all logarithmic tables.

6 No. 53— USE OF LOGARITHMS

in the left-hand column, and in the column under "6" at the top of the page. The figures are ^049. The sign # indicates that the two figures to be prefixed are not 65, as would ordinarily be the case, but 66, or the figures given in the next following line in the 0-column. This rule should always be borne in mind. Hence, log 4.576 = 0.66049.

Example 5. — Find log 72.

To find the mantissa, proceed as if it were to be found for 7200. This we find from the tables to be .85733. The characteristic of the logarithm of 72 is 1. Hence log 72 = 1.85733.

Example 6. — Find log 0.007631.

To find the mantissa, proceed as if it were to be found for 7631. This we find from the tables to be .88258. The characteristic is 3, according to the rule given for characteristics of logarithms of num- bers less than 1. Hence, log 0.007631 = 3-88258. i

Example 7. — Find log 37,262.

While we will later explain how to find more exactly the mantissa for a number with five figures, at present we may consider it accurate enough for our purpose to find the mantissa for four figures, or for 3726. This is .57124. The characteristic of the logarithm of 37,262 is 4. Hence log 37,262 = 4.57124. This, of course, is only an approxima- tion, but is near enough for nearly all shop and general engineer- ing calculations.

If the given number had been 37,267 instead of 37,262, the logarithm should have been found for 3727, as the fourth figure then should have been increased by 1, when dropping the fifth figure, which is larger than o.

Below are given several examples of numbers with their logarithms. A careful study of these examples, the student finding the logarithms for himself from the tables, and checking them with the results given, will tend to make the methods employed clearer and fix them in the

mincl.

Number Logarithm

16.95 1.22917

2 0.30103

966.2 2.98507

151 2.17898

3.5671 0.55230

12.91 1.11093

3803.8 3.58024 /^

0.007 3.84510

It should be understood that in logarithms of numbers less than 1, the characteristic, only, is negative. The mantissa is always positive, so that 3.84510 actually means (— 3) + 0.84510.

Finding the Number whose Logarithm is Given

When a logarithm is given, and it is^ required to find the corres- ponding number, first find the first two figures of the mantissa in the column headed "0" in the tables. Then find in the group of mantissas, all having the same first two figures, the remaining three figures.

METHODS, RULES AND EXAMPLES 7

These may be in any of the columns headed "0" to "9." The number heading the column in which the last three figures of the mantissa were found, is the last figure in the number sought, and the number in the left-hand column, headed "N," in line with the figures of the mantissa, gives the three first figures in the number sought.

,When the actual figures in the number sought have thus been deter- mined, locate the decimal point according to the rules given for the characteristic of logarithms. If the characteristic is greater than 3, ciphers are added. For example, if the figures corresponding to a certain mantissa are 3765, and the characteristic is 5, then the num- ber sought must have 6 figures to the left of the decimal point, and ^ \n hence would be 376500. If the characteristic had been 3, then the •£* -- number sought, in this case, would have been 0.003765.

If the mantissa is not exactly obtainable in the tables, find the near- v( est mantissa in the table to the one given, and determine the number corresponding to this. In most cases this gives ample accuracy. A method will be explained later whereby still greater accuracy may be obtained, but for the present it will be assumed that the numbers corresponding to the nearest mantissa in the tables are accurate enough for practical purposes.

A few examples will now be given in which it is required to find the number when the logarithm is given.

Example 1. — Find the number whose logarithm is 3.89382.

First find the firs.t two figures of the mantissa (89) in the column headed "0" in the tables. Then find the remaining three figures (382) in the mantissas which all have 89 for their first two figures. The figures "382^ are found in the column headed "1," which thus is the last figure in the number sought; the figures "382" are also opposite the number 783 in the left-hand column, which gives the first three figures in the number sought. The figures in this number, thus, are 7831, and as the characteristic is 3, it indicates that there are four figures to the left of the decimal point, or, in other words, that 7831 is a whole number.

Example 2. — Find the number whose logarithm is 2.75020.

First find the first two figures of the mantissa (75) in the column head "0" in the tables. Then find the remaining three figures (020) in the mantissas, which all have 75 for their first two figures. The # in front of the figure ^020 in the line next above that in which 75 was found indicates that these figures belong to the group preceded by 75. Therefore, as ^.020 is found in the column headed "6" and opposite the number 562 in the left-hand column, the figures in the number required to be found are 5626. As the characteristic is 2, th«e decimal point is placed after the first three figures, and, hence, the number whose logarithm is 2.75020 is 562.6.

Example 3. — Find the number whose logarithm is 2.45350.

After having located 45 iif the column headed "0," it will be found that the last three figures (350) of the mantissa are not to be found in the table in the group preceded by 45. The nearest value in the table, which is 347, is, therefore, located, and the corresponding num-

8 No. 53— USE OF LOGARITHMS

ber is found to be 284.1, the decimal point being placed after the third figure, because the characteristic of the logarithm is 2. Had the char- acteristic of the logarithm been 5 instead of 2, the number to be feund would have been 284,100.

Below are given a selection of examples of logarithms with tbybir corresponding numbers. The student should find the numbers/for himself from the tables, and check them with the results "given. This will aid in fixing the rules and methods employed more firmly in the mind.

Corresponding Logarithm Number

1.43201 27.04

4.89170 77,930

2.76057 0.05762

0.12096 1.321

2.99099 979.5

T.60206 0.4

5.60206 400,000

It being now assumed that the student has mastered the methods for finding the logarithms for given numbers, and the numbers for given logarithms, from the tables, the use of logarithms in multipli cation and division will next be explained.

Multiplication by Logarithms

// two or more numbers are to be multiplied' together, find the logarithms of the numbers to be multiplied, and then add these logarithms; the sum is the logarithm of the product, and the numbed corresponding to this logarithm is the required product.

Example 1. — Find the product of 2831 X 2.692 X 29.69 X 19.4. This calculation is carried out by means of logarithms as follows: log 2831. =3.45194 log 2.692 = 0.43008 log 29.69 =1.47261 log 19.4 =1.28780

6.64243

The sum of the logarithms, 6.64243, is the logarithm of the product, and from the tables we then find that the product equals 4.390,000. This result is, of course, only approximately correct, at the last three figures are added ciphers; but for most engineering calculations the result would give all the accuracy required. In most engineering calculations one or more factors are assumed from experimental values, and as these assumed values evidently must often vary be- tween wide limits, it would show lack of judgment to require calcula- tions in which such assumed values enter, to be carried out with too many "significant" figures. Such values are fully as well expressed in round numbers, with ciphers annexed to give the required value to the figures found from the tables.

If one or more of the characteristics of the logarithms are negative, these are subtracted instead of added to the sum of the character-

METHODS, RULES AND EXAMPLES 9

istics. The mantissas, as already mentioned, are always positive, so that they are always added in the usual manner. In order to fully understand the adding of positive and negative numbers in the following examples, the student should be* familiar with calculations with positive and negative quantities, as explained in MACHINERY'S. Reference Series No. 54, Solution of Triangles, Chapter III. Example 2. — Find the product 371.2 X 0.0972 X 3.

log 371.2 =2.56961

log 0.0972=2.98767 7. *>

log 3. = 0.47712

2.03440

The number corresponding to the logarithm 2.03440 is 108.2. Note that the first two figures of the mantissa of the logarithm are 03. Example 3. — Find the product 12.76 X 0.012 X 0.6. log 12.76 =1.10585 log 0.012=2.07918 7 log 0.6 =1.77815 "

2.96318 The product, hence, is 0.09187.

Division by Logarithms

When dividing one number by another, the logarithm of the divisor is subtracted from the logarithm of the dividend. The remainder is the logarithm of the quotient.

For example, if we are to find the quotient of 7568 -r- 935.3, we first find log 7568 and then subtract from it log 935.3. The remainder is then the logarithm of the quotient.

It is advisable, however, to make a modification, as explained in the following, of the logarithm of the divisor so as to permit of its addition to, instead of its subtraction from the logarithm of the divi- dend. Assume, for instance, that an example, as below, were given: 375.2 X 97.2 X 0.0762 X 3

962.1 X 92 X 33.26

It would be perfectly correct to find the logarithms of all the fac- tors in the numerator and add them together, and then the logarithms of all the factors in the denominator and add them together; and finally subtract the sum of the logarithms of the denominator from the sum of the logarithms of the numerator. The remainder is the logarithm of the result of the calculation. This method, however, involves two separate additions and one subtraction. It is possible, by a modification of the logarithms of the numbers in the denominator to so arrange the calculation that a single addition will give the logarithm of the final result.

In dealing with positive and negative numbers we learn that if we add a negative number to a positive number, the sum will be the same as if we subtract the numerical value of the negative

10 No. 53— USE OF LOGARITHMS

number from the positive number; that is 5 -f- ( — 2) = 5 — 2 •= 3. If we reverse this proposition ve have 5 — 2 = 5 -f ( — 2). If we now assume that 5 is the logarithm of a certain number a and 2 the logarithm of another number Z>, and if we insert these values in the last expression, instead of 5 and 2, we have:

log a — log 6 = log a 4- (— log &).

From this we see that instead of subtracting log & from log a we can add the negative value of log 6 and obtain the same result. As the mantissa always must remain positive, in order to permit direct addition, the negative value of the logarithm cannot be obtained by simply placing a minus sign before it. Instead, it is obtained in the following manner:

If the characteristic is positive, add 1 to its numerical value and place a minus sign over it. To obtain the mantissa, subtract the given mantissa from 1.00000.

Example 1. — The logarithm of 950 = 2.97772. Find ( — log 950).

According to the rule given, the characteristic will be 3. The man- tissa will be 1.00000 — .97772 = .02228. The last calculation can be carried out mentally without writing it down at all, by simply finding the figure which, added to the last figure in the given mantissa would make the sum 10, and the figures which added to each of the other figures in the mantissa, would make the sum 9. as shown below:

97772 02228

9 9 9 9 10

As this calculation is easily carried out mentally, the method described, when fully mastered, greatly simplifies the vrork where operations of both multiplication and division are to be performed in the same example.

Example 2.— The logarithm of 2 is 0.30103. Find (— log 2).

According to the given rules the characteristic is T, and the mantissa, .69897.

The following examples should be studied until thoroughly under- stood:

log 270. =2.43136 - log 270. =3.56864

log 10. =1.00000 — log 10. = T.OOOOO

log 26.99 =1.43120 - log 26.99 " = 5.56880

In the example in the second line an exception from the rule for obtaining the mantissa of the negative logarithm is made. It is obvious, however, that if log 10 = 1.00000, then ( — log 10) = T.OOOOO. In the example in the last line there is another deviation from the literal understanding of the rule for the mantissa. As the last figure in the positive logarithm is 0, the last figure in ( — log 26.99) is also 0, and the next last figure is treated as if it were the last, making the next last figure in the negative logarithm 8. ,

If the characteristic of the logarithm is negative, subtract 1 its numerical value, and make it positive. The mantissa is obtained by the same rule as before.

METHODS. RULES AND EXAMPLES

11

Example 1.— The logarithm of 0.003 = 3.47712. Find ( — log 0.003). According to the rule just given the characteristic will be 2. The mantissa will be .52288. Hence (— log 0.003) = 2.52288.

The following examples should be studied until fully understood: log 0.3 — T.47712 —log 0.3 =0.52288

log 0.0006963=3.84280 -log 0.0006963 =3.15720

log 0.6607 = 1.82000 —log 0.6607 =0.18000

When sufficient practice has been obtained, the negative value of a logarithm can be read off almost as quickly from the tables as the positive value given, and the subsequent gain of time, and the ease of the calculations following, more than justify this short-cut method.

Examples of the Use of Logarithms

We will now give a number of examples of the use of logarithms in calculations involving multiplication and division. No comments will be made, as it is assumed that the student has now grasped the principles sufficiently to be able to follow the methods used without further explanation.

Example 1.

0.0272 X 27.1 X 12.6.

2.371 X ,0.007 log 0.0272 = 2.43457 log 27.1 =1.43297 12.6 =1.10037 2.371 =T.€2507 0.007 =2.15490

log

— log — log

2.74788 The result, then, is 559.6.

Example 2.

0.3752 X 0.063 X 0.012

0.092 X 1289

log 0.3752 = T.57426

log 0.063 =2.79934

log 0.012 =5.07918

-log 0.092 =1.03621

— log 1289.0 =4.88975

The result, then, is 0.000002392. Example 3.

6.37874

3.463 X 1.056 X 14.7 X 144 X 10

log log log

log log

The result, then, is 77,410.

3.463 = 0.53945

1.056 = 0.02366

14.7 =i:i6732

144.0 10.0

= 2'.15836 = 1.00000

4.88879

12 No. 53— USE OF LOGARITHMS

Example 4.

0.00005427 X 392 X 2.5 X 200 X 200 log 0.00005427 = 5.73456

log 392. —2.59329

log 2.5 =0.39794

log 200. =2.30103

log 200. =2.30103

3.32785 Hence, the result is 2127.

Obtaining: the Powers of Numbers

Expressions of the form 6.513 can easily be calculated by means of logarithms. The small (3) is called exponent.* In this case the "third power" of 6.51 is required.

A number may be raised to any power by simply multiplying the logarithm of the number by the exponent of the number. The product gives the logarithm of the value of the power.

Example 1. — Find the value of 6.51s.

log 6.51 = 0.81358 3 X 0.81358 = 2.44074

The logarithm 2.44074 is then the logarithm of 6.513. Hence 6.51* equals the number corresponding to this logarithm, as found from the tables, or 6.513 = 275.9. «

Example 2. — Find the value of 12 1-29.

log 12 = 1.07918 1.29 X 1.07918 = 1.39214

Hence, 12 1-29 = 24.67.

The multiplication 1.29 x 1.07918 is carried out in the usual arith- metical way. The example above is one of a type which cannot be solved by any means except by the use of logarithms. An expression of the form 6.513 can be found by arithmetic by multiplying 6.51 X 6.51 X 6.51, but an expression of the form 121-29 does not permit of being calculated by any arithmetical method. Logarithms are here absolutely essential.

One difficulty is met with when raising a number less than 1 to a given power. The logarithm is then composed of a negative term, the, characteristic, and a positive term, the mantissa. For example: Find the value 0.313. The logarithm of 0.31 = T.49136. In this case, multi- ply, separately, the characteristic and the mantissa by the exponent, as shown below. Then add the products.

log 0.31 =(1149136

V — JV^ — ^

Multiplying characteristic and mantissa separately by 5 we have:

5 Xl = 5 5 X .49136 = 2.45680

log 0.315 = 3.45680 Hence, 0.315 = 0.002863.

* See MACHINERY'S Reference Series No. 52,, Advanced Shop Arithmetic for the Machinist, Chapter III.

METHODS, RULES AND EXAMPLES 13

If the exponent is not a whole number, the procedure will be some- what more complicated. The principle of the method, however, re- mains the same.

Example: Find the value of 0.062 -31

log 0.06 = 5.77815 Then

2.31 X" 2 = 2.31 X (—2) = — 4.62

2.31 X 0.77815 = 1.79753

In this case, tire first product, — 4.62, is negative both as regards the whole number and the decimal. In order to make the decimal positive so that we may be able to add it directly to the second product, 1.79753, we must use the same rule as given for changing a logarithm with a positive characteristic to a negative value. Hence — 4.62 = 5.38. We can now add the products:

5.38 1.79753

log 0.062-31 =S.17753 Hence 0.062-81 = 0.001505. As a further example, find 0.073-51.

log 0.07 = 5.84510 Then

3.51 X 5 = 3.51 X (—2)= — 7.02=3.98

3.51 X .84510 =2.96630

log 0.073-51= 5.94630 Hence 0.073-51 =0.00008837.

Extracting- Roots by Logarithms

Roots of numbers, as for example i/O$T, can easily be extracted by means of logarithms. The small (5) in the radical (V) of the root- sign is called the index of the root. In the case of the square root the index is (2), but it is not usually indicated, the square root being merely expressed by the sign V.

Any root of a number may be found by dividing its logarithm by the index of the root; the quotient is the logarithm of the root.

Example 1. — Find -^ 276.

log 276 = 2.44091 2.44091 -=-3 = 0.81364

Hence log f 7~276~= 6.81364, and ^"276"= 6.511.

Example 2.— Find -^KiuJTT | j

log 0.67 = 1.82607

In this case we cannot divide directly, because we have a negative characteristic and a positive mantissa. We then proceed as follows: Add numerically as many negative units or parts of units to the character- istic as is necessary to make it evenly contain the index of the root. Then add the same number of positive units or parts of units to the mantissa. Divide each separately by the index. The quotients give

14 No. 53— USE OF LOGARITHMS

the characteristic and mantissa, respectively, of the logarithm of the root.

Proceeding with the example above according to this rule, we have:

1 + 2 = 3; 3 + 8 = I.

.82607 + 2 = 2.82607; 2.82607 -*- 3 = .94202.

Hence, log f/lK67 = T.94202, and ^ 0.67 = 0.875.

Example 3.— Find V °-2-

log 0.2 = 1.30103.

If we add ( — 0.7) to the characteristic of the logarithm found, it will be evenly divisible by the index of the root.

Hence:

T + (—0.7) = —1.7; —1.7-^-1.7 = 1. .30103 + 0.7 = 1.00103 ; 1.00103 -=- 1.7 = .58884.

Hence, log ™/~03 = T.58884, and l] ~OJJ = 0.388. A number of examples of the use of logarithms in the solution of everyday problems in mechanics, are given in MACHINERY'S Reference Series No. 19, Use of Formulas in Mechanics, Chapter II, 2nd edition.

When exponents or indices are given in common fractions, it is usually best to change them to decimal fractions before proceeding further with the problem.

Interpolation

If the number for which the logarithm is required consists of five figures, it is possible, by means of the small tables in the right-hand column of the logarithm tables, headed "P. P." (proportional parts), to obtain the logarithm more accurately than by taking the nearest value for four figures, as has previously been done in the examples given. The method by which the logarithm is then obtained is called interpolation.

In the same way, if a logarithm is given, the exact value of which cannot be found in the tables, the number corresponding to the logar- ithm can be found to five figures by interpolation, although the main tables contain only numbers of four figures.

The logarithm of 2853 is 3.45530, and the logarithm of 2854 is 3.45545, as found from the tables. Assume that the logarithm of 2853.6 were required. It is evident that the logarithm of this latter number must have a value between the logarithms of 2853 and 2854. It must be somewhat greater than the logarithm of the former number, and somewhat smaller than that of the latter. While the logarithms, in ceneral, are not proportional to the numbers to which they corres- pond, the difference is very slight in cases where the increase in the numbers is small; so that, in the case of an increase from 2853 to 2854, the logarithms for the decimals 2853.1, 2853.2, etc., may be considered proportional to the numbers. It is on this basis that the small tables in the right-hand column headed "P.P." are calculated, and the logarithm of 2853.6, for example, is found as follows:

Find first the difference between the nearest larger and the nearest smaller logarithms. Log 2854 = 3.45545 and log 2853 = 3.45530. The

METHODS, RULES AND EXAMPLES 15

difference is 0.00015. Then in the small table headed "15" in the right- hand column find the figure opposite 6 (6 being the last or fifth figure in the given number). This figure is 9.0. Add this to the mantissa of the smaller of the two logarithms already found, disregarding the decimal point in the mantissa, and considering it, for the while being as a whole number. Then 45530 -f 9.0 = 45539. This is the mantissa of the logarithm of 2853.6, and the complete logarithm is 3.45539.

Example. — Find log 236.24.

Log 236.2 = 2.37328; log 236.3 = 2.37346; difference — 0.00018. In table "18" the proportional part opposite 4 is 7.2. Then 37328 + 7.2 = 37335.2. The decimal 2 is not used, but is dropped. Hence log 236.24 = 2.37335.

If the proportional part to be added has a decimal larger than 5, it should not be dropped before the figure preceding it has been raised one unit. For example, if the logarithm of 236.26 had been required, then the proportional part would have been 10.8 and the mantissa sought 37328 + 10.8 = 37338.8. Now the decimal 8 cannot be dropped before the figure 8 preceding it has been raised to 9. Then log 236.26 = 2.37339.

If the number for which the lorgarithm is to be found consists of more than five figures, find the mantissa for the nearest number of five figures, but choose the characteristic according to the total num- ber of figures to the left of the decimal point. For example, if the logarithm of 626,923 is required, find the mantissa, by interpolation, for 62692. If the logarithm for 626,928 is required, find the mantissa for 62693, always remembering to raise the value of the last figure, if the figure dropped is more than 5. The characteristic in each of these examples would, of course, be 5, as it is chosen according to the total number of figures to the left of the decimal point in the given numbers, which is 6.

To find a number whose logarithm is given more accurately than to four figures, when the given mantissa cannot be found exactly in the tables, find the mantissa which is nearest to, but less than the given mantissa. Subtract this mantissa from the nearest larger mantissa in the tables and find in the right-hand column the small table headed by this difference. Then subtract the nearest smaller mantissa from the given logarithm, and find the difference, exact or approximate, in the "proportional part" table (in the right-hand column of this table). The corresponding figure in the left-hand column of the "pro- portional part" table is the fifth figure in the number sought, the other four figures being those corresponding to the logarithm next mailer to the given logarithm.

Example. — Find the number whose logarithm is 4.46262.

The mantissa can not be found exactly in the tables; therefore, fol- lowing the rules just given, we see that the nearest smaller mantissa in the tables equals 46255. The next larger is 46270. The difference between them is 15. The difference between the mantissa of the given logarithm, 46262 and the next smaller mantissa, 46255 is 7. Now, in the proportional parts table opposite 7.5 in the right-hand column of

16 No. 53— USE OF LOGARITHMS

the table headed 15, we find that the fifth figure of the number sought would be 5. The four first figures are 2901. Hence the number sought is 29,015.

The following examples, if carefully studied, will give the student a clear conception of the method of interpolation.

Number Logarithm

52,163 4.71736

26.913 1.42996

0.012635 5.10157

12.375 1.09254

6.9592 0.84256

The student should find for himself, first the logarithms correspond- ing to the given numbers, and then the numbers corresponding to the given logarithms. In this way a check on the accuracy of the work can be obtained by comparing with the results given.

General Remarks

In the system of logarithms tabulated on pages 18 to 35, the base of the logarithms is 10; that is, the logarithm is actually the exponent which would be affixed to 10 in order to give the number correspond- ing to the logarithm. For example log 20 = 1.30103, which is the same as to say that 101-30103 = 20. Log 100 = 2, and, of course, we know that 102 = 100. As 101 = 10, the logarithm of 10 = 1. The logarithm of 1 = 0. The system of logarithms having 10 for its base is called the Briggs or the common system of logarithms.

"While the accompanying logarithm tables are given to five decimals, it should be understood that the logarithm of a number can be calcu- lated with any degree of accuracy, so that large logarithm tables give the logarithm with as many as seven decimal places, and some, used for very accurate scientific investigations, give as many as ten deci- mals. It will be noticed that in the accompanying tables the figure 5, when in the fifth decimal place, is either written 5 or 5. If the sixth place is 5 or more, the next larger number is used in the fifth place, and the logarithm is then written in the form 3.90855. The dash over the 5 shows that the logarithm is less than given. If the sixth figure is less than 5, the logarithm is written 3.91025, the dot over the 5 showing that the logarithm is more than given. In calcu- lations of the type previously explained, this, however, need not be taken into consideration and these signs should be disregarded by the student.

Hyperbolic Logarithms

In certain mechanical calculations, notably those involving the calcu- lation of the mean effective pressure of steam in engine cylinders, use is made of logarithms having for their base the number 2.7183, com- monly designated e, and found by abstract mathematical analysis. These logarithms are termed hyperbolic, Napierian or natural; the preferable name, and that most commonly in use in the United States is hyperbolic logarithms. The hyperbolic logarithms are usually desig- nated "hyp. log." Thus, when log 12 is required, it always refers to

METHODS, RULES AND EXAMPLES 17

common logarithms, but when the hyp. log 12 is required, reference is made to hyperbolic logarithms. Sometimes, the hyperbolic loga- rithm is also designated "loge" and "nat. log."

To convert the common logarithms to hyperbolic logarithms, the former should be multiplied by 2.30258. To convert hyperbolic loga- rithms to common logarithms, multiply by 0.43429. These multipliers will be found of value in cases where hyperbolic logarithms are re- quired in formulas. Hyperbolic logarithms find extensive use in higher mathematics.

SECTION II

TABLES OF COMMON LOGARITHMS

1 TO 10,000

18

No. 53— USE OF LOGARITHMS

H.	L. 0 1 2 3 4	56789	P.P.
100 101 102 103 104	oo ooo 043 087 130 173 432 475 5l8 561 6<H 860 903 945 988 #030 01 284 326 368 410 452 703 745 787 828 870	217 260 303 346 389 647 689 732 775 817 #072 #115 *i57 #199 #242 494 536 578 620 662 912 953 225 #036 #078	i 2 3 4 I 9 i 2 3 4 7 8 9 i 2 3 4 i 9 i 2 3 4 y S 9 i 2 3 4 i 9	44 43 42 4,4 4,3 4,2 8,8 8,6 8,4 13,2 12,9 I2,6 17,6 17,2 16,8 22,0 21,5 2I,° 26,4 25,8 25,2 30,8 30,1 29,4 35,2 34,4 33,6 39.6 38,7 37,8 41 40 39 •4,1 4,o 3,9 8,2 8,0 7,8 12,3 I2,° IJ,7 16,4 16,0 15,6 20,5 20,0 19,5 24,6 24,0 23,4 28,7 28,0 27,3 32,8 32,0 31,2 36,9 36,0 35,1 38 37 36 3,8 3,7 3,6 7,6 7,4 7,2 11,4 II,1 Io,8 15,2 14,8 14,4 19,0 18,5 18,0 22,8 22,2 21,6 26,6 25,9 25,2 30,4 29,6 28,8 34,2 33,3 32,4 35 34 33 3,5 3-4 3,3 7,0 6,8 6,6 10,5 10,2 9,9 14,0 13,6 13,2 17,5 17,0 16,5 2I,O 2O,4 19,8 24,5 23,8 23,I 28,0 27,2 26,4 31,5 30,6 29,7 32 31 30 3,2 3,1 3,0 6,4 6,2 6,0 9,6 9,3 9,o 12,8 12,4 I2,° 16,0 15,5 15,0 19,2 18,6 18,0 22,4 2I,7 21,0 25,6 24,8 24,0 28,8 27,9 27,0
107 108 109	02 119 l6o 202 243 284 531 572 612 653 694 938 979 *oi9 *o6o #ioo 03342 383 423 463 503 743 782 822 862 902	325 366 407 449 490 735 776 816 857 898 #141 #181 #222 ^262 ^302 543 583 623 663 703 941 981 *O2I #060 #IOO
110 in 112 "3 114	04 139 179 218 258 297 532 571 6 10 650 689 922 961 999 #038 *077 05 308 346 385 423 461 690 729 767 805 843	336 376 415 454 493 727 766 805 844 883 #115 #154 ^192 #231 ,269 500 538 576 614 6^2 88z 918 956 994 *Q32
"5 116 117 118 119	06 070T1J08 145 183 221 446 483 521 558 595 819 856 893 930 967 07.188 225 262 298 .335 555 59i 628 664 700	258 296 333 371 408 633 670 707 744 781 *oo4 #041 #078 .»ii5 #151 372 408 445 482 518 737 773 809 846 882
120 121 122 123 124	918 954 990 #027 #063 08 279 314 350 386 422 636 672 707 743 778 991 #026 *o6i ^096 #132 09 342 377 412 447 482	*099 *I35 #171 *207 *243 458 493 529 565 600 814 849 884 920 955 #167 *202 #237 #272 ^307 5J7 552 587 621 656
125 126 127 128 129	691 726 760 795 830 10 037 072 106 140 175 380 415 449 483 517 721 755 789 823 857 ii 059 093 126 160 193	864 899 934 968 *oo3 209 243 278 312 346 551 585 619 653 687 890 924 958 992 #025 227 261 294 327 361
130 I31 132 133 134	394 428 461 494 528 727 760 793 826 860 12057 090 123 156 189 385 418 450 483 516 710 743 775 808 840	561 594 628 661 ,694 893 926 959 902 #024 222 254 287 360 352 548 581 613 646 678 872 905 937 969 *ooi
135 136 137 138 139	13033 066 098 130 162 354 386 418 450 481 672 704 735 767 799 988 #019 ^051 #082 *H4 H30I 333 304 395 426	194 226 258 290 322 5J3 545 577 609 640 830 862 893 925 956 #145 #176 #208 #239 #270 457 489 520 551 582
140 141 142 143 144	613 644 675 706 737 922 953 983 #014 #045 15229 259 290 320 351- 534 564 594 625 655 836 866 897 927 957	768 799 829 860 891 #076 #106 #137 #108 #198 —381 412 442 473 503 685 7i5 746 776 806 987 #017 #047 #077 *I07
$ 147 148 149	16 137 167 197 227 256 435 465 49? 524 554 732 761 791 820 8fo 17026 056 085 114 143 3J9 348 377 406 435	286 316 346 376 406 584 613 643 673 702 879 909 938 967 997 173 202 231 260 289 464 493 522 551 580
150	609 638 667 696 725	754 782 811 840 869
N.	L. 0 1 2 3 4	56789	p.p.

LOGARITHMIC TABLES

19

N.	L. 0 1 2 3 4	56789		P.P.
150 152	17 609 638 667 696 725 898 926 955 984 *oi3 18 184 213 241 270 298 469 498 526 554 583 752 780 808 837 865	754 782 811 840 869 327 355 384 412 441 611 639 667 696 724 893 921 949 977 *005	i 2 3	29 28 2,9 2,8 5,8 5,6 8/7 8,4
156 158 159	19033 061 089 117 14^ 312 340 368 396 424 590 618 645 673 700 866 893 921 948 976 20 140 167 194 222 249	173 201 229 257 285 451 479 507 535 562 728 756 783 811 838 ^003 ^030 $058 ^085 #112 276 303 330 358 385	4 1 9	11,6 11,2 14,5 14,0 17,4 16,8 20,3 19,6 23,2 22,4 26,1 25,2
160 161 162 163 164	412 439 466 493 520 683 710 737 763 790 952' 978 *oo5 ^032 ^059 21 219 245 272 299 325 484 511 537 564 590	54s 575 602 629 656 817 844 871 898 925 352 378 405 431 458 617 643 669 696 722	I 2 3	27 26 2,7 2,6 5,4 5,2 8,1 7,8
167 168 169	748 775 801 827 854 22 01 1 037 063 089 115 272 298 324 350 376 531 557 583 608 634 789 814 840 866 891	880 906 932 958 985 141 167 194 220 246 401 427 453 479 505 660 686 712 737 763 917 943 968 994 *oi9	4 I 9	10,8 10,4 J3,5 J3,o 16,2 15,6 18,9 18,2 21,6 2O,8 24,3 23,4
170 171 172 173 174.	23045 070 096 121 147 300 325 350 376 401 553 578 603 629 654 805 830 855 880 905 24055 080 105 130 155	172 198 223 249 274 426 452 477 502 528 679 704 729 754 779 930 953 98o *oo5 ^030 i 80 204 229 254 279		25 i 2,5 3 7.5
175 176 177 178 179	304 329 353 378 403 551 576 601 625 650 797 822 846 871 895 25 042 066 091 115 139 285 3io 334 358 382	428 452 477 502 527 674 699 724 748 773 920 944 969 993 *oi8 164 188 212 237 261 406 431 455 479 503		4 I0,° 5 12,5 6 15,0 7 17,5 8 20,0 9 22,5
180 181 182 183 184	527 55i 575 600 624 768 792 816 840 864 26 007 031 055 079 102 245 269 293 316 340 482 505 529 553 576	648 672 696 720 744 888 912 935 959 983 126 I^O 174 198 221 364 387 4ii 433 458 600 623 647 670 694	i 2	24 23 2,4 2,3 4,8 4,6 7,2 6,9
185 186 187 188 189	717 741 7631 788 811 951 973 998 *02i *045 27 184 207 231 254 277 416 439 462 485 508 646 609 692 715 738,	834 858 88 i 905 928 300 323 346 370 393 531 554 577 600 623 761 784 807 830 852	4 1 9	9/6 9,2 12,0 11,5 16*8 16^1 19,2 18,4 21,6 20,7
190 191 192 193 194	875 '898 921 944 967 28 103 126 149 171 194 .33° 353 375 398 421. 556 ^78 601 623 646 780 803 825 847 870	989 #012 $035 #058 #08 1 217 240 262 285 307 443 466 488 511 533 668 691 713 735 758 892 914 937 959 981	i 2 3	22 21 2/2 2,1 4/4 4/2 6/6 6,3
195 196 197 198 199	29 003 026 048 070 092 226 248 270 292 314 447 469 49i 513 533 667 688 710 732 754 885 907 929 951 973	115 137 159 181 203 336 358 380 403 423 557 579 601 623 645 776 798 820 842 863 994 *oi6 #038 *o6o *o8i	4 7 8 9	8/8 8,4 11,0 10,5 13,2 12,6 15,4 i4,7 17,6 16,8 19,8 18,9
200	3q 103 125 146 168 190	211 233 255 276 298
N.	L. 0 1 2 3 4	56789		P.P.

20

No. 53— USE OF LOGARITHMS

N.	L. 0 1 2 3 4	56789	P.P.
200 2OI 2O2 203 204	30 103. 125 146 168 190 320 341 363 384 406 535 557 578 600 621 750 771 792 814 835 963 984 *oo6 #027 ^048	211 233 255 276 298. 428 449 471 492 514 643 664 685 707 728 856 878 899 920 942 #069 #091 *H2 *I33 *I54	22 I 2, 2 4 3! 6, 4 8 5 ii, 6 13, 7 15, 8 17, 9li9, I 2 3 4 9 I 2 3 4 5 6 7 8 9 I 2 3 4 7 8 9 I 2 3 4 7 8 9	21 2 2,1 4 4,2 6 6,3 8 8,4 0 I0,5 2 12,6 4 14,7 6 16,8 8 18,9 20 2,0 4,o 6,0 8,0 IO,O I2,O 14,0 16,0 18,0 19 1,9 3,8 % 9,5 ",4 13,3 15,2 17,1 18 1,8 3,6 5,4 7,2 9,o 10,8 12,6 14,4 16,2 17 i,7 3,4 5,1 6,8 8,5 10,2 n,9 13,6 15,3
205 206 207 208 209	31 175 197 218 239 260 387 4Q§ 429 450 471 597 618 639 660 681 806 827 848 869 890 32015 035 056 077 098	281 302 323 345 366 492 5*3 534 555 576 702 723 744 765 785 911 931 952 973 994 118 139 100 181 201
210 211 212 213 214	222 243 263 284 305 428 449 469 490 510 634 654 675 695 715 838 858 879 899 919 33 041 062 082 IO2 122	325 346 366 387 408 53i 552 572 593 613 736 756 777 797 818 940 960 980 #00 i #021 143 163 183 203 224
215 216 217 218 219	244 264 284 304 325 445 465 486 506 526 646 666 686 706 726 846 866 885 905 925 34044 064 084 104 124	345 365 385 405 425 546 566 586 606 626 746 766 786 806 826 945 96$ 985 *oo5 *025 143 163 183 203 223
220 221 222 223 224	242 262 282 301 321 439 459 479 498 518 63$ 655 674 694 713 830 850 869 889 908 35 025 044 064 083 102	341 361 380 400 420 537 557 577 596 616 733 753 772 792 811 928 947 967 986 *cx>5 122 141 160 180 199
22| 226 227 228 229	218 238 257 276 295 411 430 449 468 488 603 622 641 660 679 793 813 832 851 870 984 *oo3 *02i ^040 #059	315 334 353 372 392 507 526 545 564 583 698 717 736 755 774 889 908 927 946 965 ^078 *097 #116 #135 *I54
230 231 232 233 234	36 173 192 211 229 248 361 380 399 418 436 549 568 586 605 624 736 754 773 791 810 922 940 959 977 996	267 286 305 324 342 455 474 493 5" 53O 642 661 680 698 717 829 847 866 884 903 *oi4 #033 ^051 ^070 #088
235 236 237 238 239	37 107 125 144 162 181 291 310 328 > 346 365 475 493 Sii 530 548 658 676 694 712 731 840 858 876 894 912	199 218 236 254 273 383 401 420 438 457 566 585 603 621 639 749 767 785 803 822 931 949 967 985 *003
240 241 242 243 244	38021 039 057 075 093 202 220 238 256 274 382 399 417 43$ 453 561 578 596 614 632 739 757 775 792 810	112 130 148 166 184 292 310 328 346 364 471 489 507 525 543 650 668 686 703 721 828 846 863 88 i 899
245 246 247 248 249	917 934 952 970 987 39094 in 129 146 164 270 287 305 322 340 445 463 480 498 515 620 637 655 672 690	*oo5 *023 #041 ^058 ^076 182 109 217 235 252 358 37$ 393 4io 428 533 55o 568 58$ 602 707 724 742 759 777
250	794 811 829 846 863	881 898 915 933 950
N.	L. 0 1 2 3 4	56789	P.P.

LOGARITHMIC TABLES

21

N.	L. 0 1 2 3 4	56789	I	.P.
250 251 252 253 254	39 794 811 829 846 863 967 985 *002 *oi9 ,,037 40 140 157 175 192 209 312 329 346 364 381 483 500 518 535 552	88 i 898 915 933 950 *054 ^071 #088 *io6 #123 226 243 261 278 295 398 415. 432 449 466 569 586 603 620 637	i 2 3	18 i 8 3> 5/4
in 3 259	654 671 688 705 722 824 841 858 875 892 993 *oio #027 *044 *o6i 41 162 179 196 212 229 330 347 363 380 397	739, 756 773 79° 807 909 926 943 960 976 #078 *095 *in *I28 *I45 246 263 280 296 313 414 430 447 464 481	4 8 9	7/2 9/o 10,8 12,6 14/4 16,2
260 261 262 263 264	497 514 53i 547 564 664 68 i 697 714 731 830 847 863 880 896 996 *oi2 #029 *04$ *o62 42 160 177 193 210 226	581 597 614 631 647 747 764 780 797 814 913 929 946 963 979 #078 *095 *m *I27 *I44 243 259 275 292 308	i 2 3	17 i/7 3/4 5/1
267 268 269	325 34i 357 374 39P 488 504 521 537 553 651 667 684 700 716 813 830 846 862 878 975 991 *oo8 *024 ^040	406 423 439 455 472 570 586 602 619 635 732 749 765 78i 797 894 911 927 943 959 #056 #072 #088 #104 #120	4 I 7 8 9	6/8 8,5 IO,2 «/9 13/6 15/3
270 271 272 273 274	43 136 152 169 185 201 •297 3J3 329 345 361 457 473 489 5°5" 52i 616 632 648 664 680 775 79i 807 823 838	217 233 249 265 281 377 393 409 425 44i 537 553 569 584 600 696 712 727 743 759 854 870 886 902 917	I 2 3	16 1/6 S/2 4/8
275 276 277 278 279	933 949 965 981 996 44091 107 122 138 154 248 264 . 279 295 311 404 420 436 .451 467 560 576 592 607 623	*oi2 #028 *044 #059 *075 170 i8£ 201 217 232 326 342 358 373 389 483 498 514 529 545 638 654 669 685 700	4 i i	6/4 8/0 9/6 IIj2 12,8 14/4
280 281 282 283 284	716 731 747 762 778 871 886 902 917 932 45 025 040 056 071; 086 17^ 194 209 225 240 332 347 ,362 378 393	793 809 824 840 855 948 963 979 994 *oio 102 117 133 148 163 255 271 286 301 317 408 423 439 454 469	i 2 3	15 i/S 3/o 4/5
31 287 288 289	484 500 515 530 545 637 652 667 ^82 697 788 803 818 834 849 939 954 969 984 *ooo 46090 105 120 135 150	561 576 591 606 621 712 728 743 758 773 864 879 894 909 924 *oi5 #030 ^045: *o6o #075 165" 180 195 210 225	4 5 -6 7 8 9	6/0 7/5 9/o 10,5 12,0 13/5
290 291 292 293 294	240 255 270 285 300 389 404 419 434 449 538 553 568 583 598 687 702 716 731 746 835 850 864 879 894	315 330 345 359 374 464 479 494 509 523 613 627 642 657 672 761 776 790 805 820 909 923 938 953 967	I 3	14 i/4 2/8 4/2
297 298 299	982 997 #012 ^026 ^041 47 129 144 159 173 188 276 290 305 319 334 422 436 451 465 480 567 582 596 611 625	#056 #070 #085 #100 *ii4 202 217 232 246 261 349 363 378 391 4<V 494 509 524 538 553 640 654 669 683 698	4 1 9	5/6 7/o 8/4 9/8 11,2 12,6
300	712 727 741 756 770	784 799 813 828 842
N.	L. 0 1 2 3 4	56789	I	». P.

22

No. 53— USE OF LOGARITHMS

N.	L. 0 1 2 3 4	56789	I	.P.
300 301 302 303 304	47712 727 741 756 770 857 871 885 900 914 48 ooi 015 029 044 058 144 159 173 187 202 287 302 316 330 344	784 799 813 828 842 929 943 958 972 986 073 087 101 116 130 216 230 244 259 273 359 373 387 401 416		15
305 306 3°7 308 309	430 444 458 473 487 572 586 601 615 629 714 728 742, 756 770 855 869 883 897 911 996 *oio #024 ^038 #052	501 5!5 530 544 558 643 657 671 686 700 785 799 813 827 841 926 940 954 968 982 #066 *o8o ^094 *io8 *I22	2 3 4	7*5 9/o
310 312 3i4	49 136 150 164 178 192 276 290 304 318 332 415 429 443 457 471 554 568 582 596 610 693 707 721 734 748	206 220 234 248 262 346 360 374 388 402 485 499 513 527 541 624 638 651 66 c; 679 762 776 790 803 817	9	12,0 13/5
315 316 3J7 3i9	831 845 859 872 886 969 982 996 *oio #024 50 106 120 133 147 161 243 256 270 284 297 379 393 406 420 433	900 914 927 941 955 #037 #051 #065 #079 *O92 174 188 202 215 229 311 325 338 352 365 447 461 474 488 501	i 2 4	14 2^8 4/2 5/6
320 321 322 323 324	515 529 542 556 569 651 664 678 691 705 786 799 813 826 840 920 934 947 961 974 51055 068 081 095 108	583 596 610 623 637 718 732 745 759 772 853 866 880 893 907 987 *ooi #014 #028 #041 121 135 148 162 175	i 7 8 9	8,4 9/8 11,2 12,6
325 326 329	i 88 202 215 228 242 322 335 348 362 375 455 468 481 495 508 587 601 614 627 640 720 733 746 759 772	255 268 282 295 308 388 402 415 428 441 521 534 548 561 574 654 667 680 693 706 786 799 812 825 838		13 2/6
330 332 333 334	851 865 878 891 904 983 996 *oo9 *022 #035 52 114 127 140 153 166 244 257 270 284 297 375 388 401 414 427	917 930 943 957 970 #048 *o6i #075 #088 #101 179 192 205 218 231 310 323 336 349 362 440 453 466 479 492	3 4 1	3/9 5/2 § 9/1 10,4
337 338 339	504 517 530 543 556 634 647 660 673 686 763 776 789 802 815 892 905 917 930 943 53020 033 046 058 071	569 582 595 608 621 699 711 724 737 750 827 840 853 866 879 956 969 982 994 #007 084 097 no 122 135	9	«,7 1 2
340 342 343 344	148 161 173 186 199 27^ 288 301 314 326 403 415 428 441 453 529 542 555 567 580 656 668 68 i 694 706	212 224 237 250 263 339 352 364 377 390 466 479 491 504 517 593 605 618 631 643 719 732 744 757 769	i 2 3 4	1,2 2,4 3,6 4,8 6,0
itl fg 349	782 794 807 820 832 908 920 933 945 958 54033 045 058 070 083 158 170 183 195 208 283 295 307 320 332	845 857 870 882 895 970 983 995 *oo8 *020 095 108 120 133 145 220 233 245 258 270 345 357 370 382 394	7 8 9	7,2 8/4 9/6 10,8
350	407 419 432 444 456	469 481 494 506 518
N.	L. 0 1 2 3 4	56789	I	'.P.

LOGARITHMIC TABLES

N.	L. 0 1 2 3 4	56789	I	.P.
350 352 353 354	54407 419 432 444 456 53i 543 555 568 580 654 667 679 691 704 777 790 802 814 827 900 913 923 937 949	469 481 494 506 518 593 603 617 630 642 716 728 741 753 765 839 851 864 876 888 962 974 986 998 *oi i		13
355 356 357 358 359	55023 035 047 060 072 143 157 169 182 194 267 279 291 303 315 388 400 413 425 437 509 522 534 546 558	084 096 108 121 133 206 218 230 242 253 328 340 352 364 376 449 461 473 485 497 570 582 594 606 618	2 3 * 7	2,6 3,9 5,2 7^8 9 *
360 361 362 363 364	630 642 654 666 678 75i 763 775 787 799 871 883 893 907 919 991 #003 #015 #027 $038 56 no 122 134 146 158	691 703 713 727 739 811 823 833 847 859 93i 943 955 967 979 #050 #062 #074 #086 #098 170 182 194 205 217	8 9	10,4 11,7
365 366 367 368 369	229 241 253 263 277 348 360 372 384 396 467 478 490 502 514 583 597 608 620 632 703 714 726 738 750	289 301 312 324 336 407 419 431 443 453 526 538 549 561 573 644 656 667 679 091 761 773 783 797 808	i 2 3 4	12 1,2 2,4 3,6 4,8
370 372 373 374	820 832 844 855 867 937 949 961 972 984 57 054 066 078 089 101 171 183 194 206 217 287 299 310 322 334	879 891 902 914 926 996 j|(Oo8 ^019 #031 #043 113 124 136 148 '159 229 241 252 264 276 345 357 368 380 392	I 9	6,0 7,2 8,4 9,6 10,8
375 376 377 378 379	403 415 426 438 449 519 530 542 553 563 634 646 657 669 680 749 761 772 784 795 864 8^5 887 898 910	461 473 484 496 507 576 588 600 6n 623 692 703 715 726 738 807 818 830 841 852 921 933 944 955 967	i	11 1,1
380 382 383 384	978 990 *OOI #013 *024 58 092 104 115 127 138 206 218 229 240 252 320 331 343 354 365 433 444 456 467 478	*O35 *O47 #058 4*070 jifOSi 149 161 172 184 193 263 274 286 297 309 377 388 399 4io 422 490 501 512 524 533	3 4 I	3,3 4,4 7,7 8,8
to to to to to CO CO CO CO OO vo oovi ONtn	546 557 569 58o 591 659 670 68 i 692 704 771 782 794 803 816 883 894 906 917 928 993 *oo6 *oi7 #028 #040	602 614^ 623 636 647 713 726" 737 749 760 827 838 850 861 872 939 95° 96i 973 984 ^051 #062 #073 ^084 #095	9	9,9 1 ft
390 39i 392 393 394	59 106 118 129 140 151 218 229 240 251 262 329 340 35i 362 373 439 450 461 472 483 530 561 572 583 594	162 173 184 195 207 273 284 295 306 318 384 395 406 417 428 494 506 517 528 539 603 616 627 638 649	i 2 3 4	1,0 2,0 3,o 4,o 5,°
395 396 397 398 399	660 671 682 693 704 770 780 791 802 813 879 890 901 912 923 988 999 *oio *02i ^032 60097 108 119 130 141	713 726 737 748 759 824 835 846 857 868 934 945 956 966 977 ^043 #054 #063 #076 *o86 152 163 173 184 195	7 8 9	6,0 7,o 8,0 9,o
400	206 217 228 239 249	260 271 282 293 304
N.	L. 0 1 2 3 4	56789	1	». P.

24

. 53— USE OF LOGARITHMS

N.	L. 0 1 2 3 4	56789	P.P.
400 401 402 403 404	60 206 217 228 239 249 314 325 336 347 358 423 433 444 455 4&6 531 54i 552 563 574 638 649 660 670 68 i	260 271 282 293 304 369 379 390 401 412 477 487 498 509 520 584 595 606 617 627 692 703 713 724 735	i 2 3 4 I 9 i 2 3 4 9 i 2 3 4 ! 8 9	11 1,1 2,2 3,3 4,4 4;I 1% 9,9 10 !,0 2,O 3,o 4,o 5,° 6,0 7,o 8,0 9,o 9 o,9 1,8 2,7 3,6 4,5 5,4 6,3 7,2 8,1
405 406 407 408 409	746 756 767 778 788 853 863 874 885 895 959 970 981 991 *002 61 066 077 087 098 109 172 183 194 204 2lf	799 810 821 831 842 906 917 927 938 949 #013 #023 #034 *045 *055 119 130 140 151 162 225 236 247 257 268
410 411 412 4i3 414	278 289 300 310 321 384 395 405 416 426 490 500 511 521 532 595 606 616 627 637 700 711 721 731 742	331 342 352 363 374 437 448 458 469 479 542 553 563 574 584 648 658 669 679 690 752 763 773 784 794
4i| 416 417 418 419	805 815 826 836 847 909 920 930 941 951 62 014 024 034 045 055 118 128 138 149 159 221 232 242 252 263	857 868 878 888 899 962 972 982 993 *oo3 066 076 086 097 107 I7O l8o 190 2OI 211 273 284 294 304 315
420 421 422 423 424	325 335 346 356 366 428 439 449 459 469 53i 542 552 562 572 634 644 655 665 67^ 737 747 757 767 778	377 387 397 408 418 480 490 5°° 511 521 583 593 603 613 624 685 696 706 716 726 788 798 808 818 829
425 426 427 428 429	839 849 859 870 880 941 951 961 972 982 63043 053 063 073 083 144 155 165 175 185 246 256 266 276 286	890 900 910 921 931 992 #002 *OI2 *022 #033 094 104 114 124 134 195 2O5 2l5 225 236 296 306 317 327 337
430 43i 432 433 434	347 357 367 377 387 448 458 468 478 488 548 558 568 579 589 649 659 669 679 689 749 759 769 779 789	397 407 417 428 438 498 508 518 528 538 599 609 619 629 639 699 709 719 729 739 799 809 819 829 839
435 436 437 438 439	849 859 869 879 889 949 959 969 979 988 64 048 058 068 078 088 147 157 167 177 187 246 256 266 276 286	899 909 919 929 939 998 #008 #018 #028 #038 098 108 118 128 137 197 207 217 227 237 296 306 316 326 335
440 441 442 443 444	345 355 365 375 385 444 454 464 473 483 542 552 562 572 582 640 650 660 670 680 738 748 758 768 777	395 404 4H 424 434 493 503 513 523 532 591 601 611 621 631 689 699 709 719 729 787 797 807 816 826
445 446 447 448 449	836 846 856 865 875 933 943 953 963 972 65 031 040 o^o 060 070 128 137 147 157 167 225 234 244 254 263	885 895 904 914 924 982 992 *002 *OII #021 079 089 099 108 118 176 186 196 205 215 273 283 292 302 312
450	321 331 341 350 360	369 379 389 398 408
N.	L. 0 1 2 3 4	56789	P.P.

LOGARITHMIC TABLES

25

N.	L. 0 1 2 3 4	56789	P.P.
450 45i 452 453 454	65321 331 341 350 360 418 427 437 447 456 514 523 533 543 552 610 619 629 639 648 706 715 725 734 744	369 379 389 398 408 466 475 485 495 504 562 571 581 591 600 658 667 677 686 696 753 763 772 782 792	i 2 3 4 i 9 i 2 3 4 7 8 9 i 2 3 4 5 6 I 9	10 1,0 2,0 3/° 4/o 5/° 6,0 7/o 8,0 9/o 9 0,9 1,8 2/7 3/6 4/5 5/4 6/3 7/2 8,1 8 0,8 1/6 2/4 3/2 4/o 4,« 5/6 6,4 7/2
455 456 457 458 459	801 811 820 830 839 896 906 916 92^ 935 992 *OOI *OII #020 ^030 66 087 096 106 115 124 l8l 191 200 210 219	849 858 868 877 887 944 954 963 973 982 *039 #049^058 *o68 *077 134 143 153 162 172 229 238 247 257 266
460 461 462 463 464	276 285 295 304 314 370 380 389 398 408 464 474 483 492 502 558 567 577 586 596 652 661 671 680 689	323 332 342 351 361 417 427 436 445 455 511 52* 530 539 549 605 614 624 633 642 699 708 717 727 736
465 466 467 468 469	74o 755 764 773 783 839 848 857 867 876 932 941 950 960 969 67025 034 043 052 062 117 127 136 145 154	792 801 811 820 829 88^ 894 904 913 922 978 987 997 *oo6 #015 071 080 089 099 108 164 173 182 191 201
470 47i 472 473 474	210 219 228 237 247 302 311 321 330 339 394 403 413 422 431 486 495 504 514 523 578 587 596 605 614	256 265 274 284 293 348 357 367 376 385 440 449 459 468 477 532 54i 550 560 569 624 633 642 651 660
475 476 477 478 479	669 679 688 697 706 761 770 779 788 797 852 861 870 879 888 943 952 961 970 979 68 034 043 052 061 070	7*£ 724 733 742 752 806 815 825 834 843 897 906 916 925 934 988 997 #006 *oi5 *024 079 088 097 106 115
480 481 482 483 484	124 133 142 151 160 215 224 233 242 251 305 314 323 332 341 395 404 413 422 431 485 494 502 511 520	169 178 187 196 205 260 269 278 287 296 35o 359 368 377 386 440 449 458 467 476 529 538 547 556 565
485 486 487 488 489	574 583 592 601 610 664 673 68 i 690 699 753 762 771 780 789 842 851 860 869 878 931 940 949 958 966	619 628 637 646 655 708 717 726 735 744 797 806 815 824 833 886 895 904 913 922 975 984 993 *oo2 *on
490 491 492 493 494	69 020 028 037 046 055 108 117 126 135 144 197 205 214 223 232 285 294 302 311 320 373 38i 390 399 408	064 073 082 090 099 152 161 170 179 188 241 249 258 267 276 329 338 346 35S 364 417 425 434 443 452
495 496 497 498 499	461 469 478 487 496 548 557 566 574 583 636 644 653 662 671 723 732 740 749 758 810 819 827 836 845	504 513 522 531 539 592 601 609 618 627 679 688 697 705 714 767 77$ 784 793 801 854 862 871 880 888
500	897 906 914 923 932	940 949 958 966 975
N.	L. 0 1 2 3 4	56789	P.P.

26

. 53— USE OF LOGARITHMS

N.	L. 0 1 2 3 4	56789	P.P.
500 501 502 5°3 504	69897 906 914 923 932 984 992 #ooi *oio *oi8- 70 070 079 088 096 105 157, 165 174 183 191 243 252 260 269 278	940 949 958 966 975 *027 ^036 *044 *053 *o62 114 122 131 140 148 200 209 217 226 234 286 295 303 312 321	9 i 0,9 2 1,8 3 2,7 4 3,6 5 4,5 6 5,4 7| 6,3 8 7,2 9! 8,1 8 i 0,8 2 1,6 3;2,4 4:3,2 5 4,o 6 4,8 7 5,6 8j6,4 9l7,2 f 7 i 0,7 2 1,4 3 2,1 4.2,8 5 3,5 6 4,2 7 4,9 8|5,6 9)6,3
507 508 509	329 338 346 355 364 415 424 432 441 449 501 509 518 526 535 586 595 603 612 621 672 680 689 697 706	372 381 389 398 406 458 467 475 484 492 544 552 561 569 578 629 638 64^ 65 s 663 714 723 731 740 749
510 5" 512 513 5i4	757 766 774 783 791 842 851 859 868 876 927 935 944 952 961 71 OI2 O2O 029 037 046 096 IO5 113 122 130	800 808 817 825 834 885 893 902 910 919 969 978 986 995 *oo3 054 063 071 079 088 139 147 155 164 172
5i5 5i6 5i7 5i8 5i9	181 189 198 206 214 265 273 282 29O 299 349 357 366 374 383 433 44i 45° 458 466 5J7 525 533 542 550	223 231 240 248 257 307 315 324 332 341 391 399 408 416 425 475 483 492 500 508 559 567 575 584 592
520 521 522 523 524	600 609 617 625 634 684 692 700 709 717 767 775 784 792 800 850 858 867 875 883 933 94i 950 958 966	642 650 659 667 675 725 734 742 750 759 809 817 825 834 842 892 900 908 917 925 975 983 99i 999 *oo8
525 526 527 528 529	72 016 024 032 041 049 099 107 115 123 132 181 189 198 206 214 263 272 280 288 296 346 354 362 370 378	057 066 074 082 090 140 148 156 165 173 222 230 239 247 255 304 313 321 329 337 387 395 4°3 4ii 419
530 53i 532 533 534	428 436 444 452 460 509 518 526 534 542 591 599 607 616 624 673 68 i 689 697 705 754 762 770 779, 787	469 477 485 493 501 550 558 567 575 583 632 640 648 656 665 713 722 730 738 746 795, 803 811 819 827
535 536 537 538 539	835 843 852 860 868 916 925 933 941 949 997 *oo6 #014 *O22 #030 73 078 086 094 102 in 159 167 175 183 191	876 884 892 900 908 957 96=; 973 981 989 ^038 ^046 *054 *o62 ^070 119 127 135 143 151 199 207 215 223 231
540 54i 542 543 544	239 247 255 263 272 320 328 336 344 352 400 408 416 424 432 480 488 496 504 512 560 568 576 584 592	280 288 296 304 312 360 368 376 384 392 440 448 456 464 472 520 528 536 544 552 600 608 616 624 632
545 546 547 548 549	640 648 656 664 672 719 727 735 743 751 799 807 815 823 830 878 886 894 902 910 957 965 973 98i 989	679 687 695 703 711 759 767 775 783 791 838 846 854 862 870 918 926 933 941 949 997 »oo5 *oi3 *020 *028
550	74 036 044 052 060 068	076 084 092 099 107
N.	L. 0 1 2 3 4	56789	P.P.

LOGARITHMIC TABLES

27

N.	L. 0 1 2 3 4	56789	P.P.
550 55i 552 553 554	74 036 044 052 060 068 115 123 131 139 147 194 202 210 218 22=; 273 280 288 296 304 35i 359 367 374 382	076 084 092 099 107 155 162 170 178 186 233 241 249 257 265 312 320 327 335 343 390 398 406 414 421	8 i o,3 2 1,6 3 2,4 4 S,2 5 4,o 6 4,8 75,6 8 6,4 9 7,2 7 I 0,7 2 1,4 3 2,1 4 2,8 5 3,5 6 4,2 7 4,9 8 5,6 9 6,3
555 556 557 558 559	429 437 445 453 461 507 5J5 523 53i 539 586 593 601 609 617 663 671 679 687 695 741 749 757 764 772	468 476 484 492 500 547 554 5°2 57° 578 624 632 640 648 656 702 710 718 726 733 780 788. 796 803 8 ii
560 56i 562 563 564	819 827 834 842 850 896 904 912 920 927 974 981 989 997 *oo5 75 051 059 066 074 082 128 136 143 151 159	858 865 873 88 i 889 935 943 950 958 966 #012 *o^ *028 *035 *043 089 097 105 113 120 166 174 182 189 197
567 568 569	20^ 213 220 228 236 282 289 297 305 312 358 366 374 38i 389 435 442 450 458. 465 511 519 526 534 542	243 251 259 266 274 320 328 335 343 351 397 404 412 420 427 473 481 488 496 504 549 557 565 572 580
570 57i 572 573 574	587 595 603 610 6i£ 664 671 679 686 094 740 747 755 762 770 815 823 831 838 846 891 899 906 914 921	626 633 641 648 656 702 709 717 724 732 778 785 793 800 808 853 861 868 876 884 929 937 944 952 959
§3 577 578 579	967 974 982 989 997 76 042 050 057 - 065 072 118 12=; 133 140 148 193 200 208 215 223 268 275 283 290 298	*oo5 *oi2 *020 #027 #035 080 087 095 103 no J5S 163 17° J78 185 230 238 245 253 260 305 313 320 328 335
580 58i 582 583 584	343 35° 358 365 373 418 425 433 440 448 492 500 507 515 522 567 574 582 589 597 641 649 656 664 671	380 388 395 403 410 455 462 470 477 485 530 537 545 552 559 604 612 619 626 634 678 686 693 701 708
585 586 587 588 589	716 723 730 738 745 790 797 805 812 819 864 871 879 886 893 938 945 953 96o 967 77 012 019 026 034 041	753 76o 768 775 782 827 834 842 849 856 901 908 916 923 930 975 982 989 997 *oo4 048 056 063 070 078
590 59i 592 593 594	°85 093 loo 107 115 159 166 173 181 188 232 240 247 254 262 305 313 320 327 335 379 386 393 4°i 408	122 129 137 144 151 195 203 210 217 225 269 276 283 291 298 342 349 357 364 371 415 422 430 437 444
597 598 599	452 459 466 474 481 525 532 539 546 554 597 605 612 619- 627 670 677 685 692 699 743 75o 757 764 772	488 495 503 510 517 561 568 576 583 590 634 641 648 656 663 706 714 721 728 735 779 786 793 801 808
600	815 822 830 837 844	851 859 866 873 880
N.	L. 0 1 2 3 4	56789	P.P.

28

No. 53— USE OF LOGARITHMS

N.	L. 0 1 2 3 4	56789	P.P.
600 601 602 603 604	77815 822 830 837 844 887 895 902 909 916 960 967 974 981 988 78 032 039 046 053 061 104 in 118 125 132	851 859 866 873 880 924 931 938 945 952 996 *oo3 #010 *oi7 *025 068 075 082 089 097 140 147 154 161 168	i 2 3 4 9 i 2 3 4 9 i 2 3 4 i 7 8 9	8 0,8 1,6 2,4 3,2 4,0 $ 6,4 7,2 7 o,7 i,4 2,1 2,8 3,5 4,2 $ 6,3 6 0,6 1,2 1,8 2,4 i:! % 5,4
607 608 609	176 183 190 197 204 247 254 262 269 276 319 326 333 340 347 390 398 405 412 419 462 469 476 483 490	211 219 226 233 240 283 290 297 305 312 355 362 369 376 383 426 433 440 447 455 497 5°4 5J2 5J9 526
610 611 612 613 614	533 540 547 554 561 604 611 618 625 633 675 682 689 696 704 746 753 76o 767 774 817 824 831 838 845	569 576 583 590 597 640 647 654 -66 I 668 711 718 725 732 739 #81 789 796 803 810 852 859 866 873 880
615 616 617 618 619	888 895 902 909 916 958 965 972 979 986 79029 036 043 050 057 099 106 113 120 127 169 176 183 190 197	923 930 937 944 951 993 *ooo #007 #014 *02i 064 071 078 085 092 134 141 148 155 162 204 211 218 22=; 232
620 621 622 623 624	239 246 253 260 267 309 316 323 330 337 379 386 393 400 407 449 456 463 470 477 518 525 532 539 546	274 281 288 295 302 344 35i 358 365 372 414 421 428 435 442 484 491 498 505 511 553 56o 567 574 58i
625 626 627 628 629	588 595 602 609 616 657 664 671 678 68$ 727 734 74i 748 754 796 803 810 817 824 865 872 879 886 893	623 630 637 644 650 692 699 706 713 720 761 768 775 782 789 831 837 844 851 858 900 906 913 920 927
630 631 632 633 634	934 941 948 955 962 80003 OI° OI7 024 030 072 079 085 092 099 140 147 154 161 168 209 216 223 229 236	969 975 982 989 996 037 044 051 058 065 106 113 120 127 134 175 182 188 195 202 243 250 257 264 271
635 636 637 638 639	277 284 291 298 305 346 353 359 366 373 414 421 428 434 441 482 489 496 502 509 550 557 564 570 577	312 318 325 332 339 380 387 393 400 407 448 455 462 468 475 516 523 530 536 543 584 591 598 604 611
640 641 642 643 644	618 625 632 638 645 686 693 699 706 713 754 760 767 774 781 821 828 835 841 848 889 895 902 909 916	652 659 665 672 679 720 726 733 740 747 787 794 801 808 814 855 862 868 875 882 922 929 936 943 949
645 646 647 648 649	956 963 969 976 983 8 1 023 030 037 043 050 090 097 104 in 117 158 164 171 178 184 224 231 238 245 251	990 996 #003 #010 #017 057 064 070 077 084 124 131 137 . 144 151 191 198 2O4 211 2l8 258 265 271 278 285
650	291 298 305 311 318	325 331 338 345 35i
N.	L. 0 1 2 3 4	56789	P.P.

LOGARITHMIC TABLES

29

N.	L. 0 1 2 3 4	56789	P.P.
650 651 652 653 654	81 291 298 305 311 318 358 365 37i 378 385 425 431 438 445 451 491 498 505 511 518 558 564 571 578 584	325 33i 338 345 35i 391 398 405 411 418 458 465 471 478 485 525 53i 538 544 55i 591 598 604 6n 617	7 I 0,7 2 1,4 3 2,1 4 2,8 5 3/5 6 4,2 7 4/9 8 5/6 9 6,3 6 i 0,6 2 1,2 3 1/8 4 2,4 5 3/o 6 3/6 7 4/2 8 4/8 9 5/4
& 656 657 658 659	624 631 637 644 651 690 697 704 710 717 757 763 770 776 783 823 829 836 842 849 889 895 902 908 915	657 664 671 677 684 723 730 737 743 75° 790 796 803 809 816 856 862 869 875 882 921 928 935 941 948
660 661 662 663 664	954 961 968 974 981 82 020 027 033 040 046 086 092 099 105 112 151 158 164 171 178 217 223 230 236 243	987 994 *ooo 3*007 *oi4 053 060 066 073 079 119 125 132 138 145 184 igi 197 204 210 249 256 263 269 276
665 666 667 668 669	282 289 295 302 308 347 354 360 367 373 413 419 426 432 439 478 484 491 497 504 543 549 556 562 569	315 321 328 334 341 380 387 393 400 406 445 452 458 465 471 510 517 523 530 536 575 582 588 595 601
670 S1 £2 3S	607 614 620 627 633 672 679 685 692 698 737 743 75° 756 763 802 808 814 821 827 866 872 879 885 892	640 646 653 659 666 705 711 718 724 730 769 776 782 789 795 834 840 847 853 860 898 905 911 918 924
%i 677 678 679	930 937 943 95o 956 995 *ooi *oo8 ^014 *02o 83059 065 072 078 085 123 129 136 142 149 187 193 200 206 213	963 969 975 982 988 *027 *033 #040 #046 3,052 091 097 104 no 117 155 161 168 174 181 219 225 232 238 245
680 681 682 683 684	251 257 264 270 276 315 321 327 334 340 378 385 39i 398 404 442 448 455 461 467 506 512 518 525 531	283 289 296 302 308 347 353 359 366 372 410 417 423 429 436 474 480 487 493 499 537 544 550 556 563
685 686 688 689	569 575 582 588 594 632 639 645 651 658 696 702 708 715 721 759 765 77i 778 784 822 828 835 841 847	601 607 613 620 626 664 670 677 683 689 727 734 740 746 753 790 797 803 809 816 853 860 866 872 879
690 691 692 693 694	885 891 897 904 910 948 954 96o 967 973 84 on 017 023 029 036 073' 080 086 092 098 136 142 148 155 161	916 923 929 935 942 979 985 992 998 *oo4 042 048 055 061 067 105 in 117 123 130 167 173 180 186 192
695 §7 698 699	198 205 211 217 223 261 267 273 280 286 323 330 336 342 348 386 392 398 404 410 448 454 460 466 473	230 236 242 248 255 292 298 305 311 317 354 361 367 373 379 417 423 429 435 442 479 485 491 497 504
700	510 516 522 528 535	54i 547 553 559 566
N.	L. 0 1 2 3 4	56789	P.P.

. 53— USE OF LOGARITHMS

N.	L. 0 1 2 3 4	56789	P.P.
700 701 702 703 704	84510 516 522 528 535 572 578 584 590 597 634 640 646 652 658 696 702 708 714 720 757 763 770 776 782	54i 547 553 559 S66 603 609 615 621 628 665 671 677 683 689 726 733 739 745 75 i 788 794 800 807 813	i 8 3 4 I I 9 i 2 3 4 7 8 9 I 2 3 4 7 8 9	7 o/7 1/4 2,1 2,8 3/5 4/2 P 6,3 6 0,6 1,2 1/8 2/4 3/o 3/6 4'o 4/8 5/4 5 o/S 1,0 i/5 2,0 2/5 3/o 3,5 4/o 4/5
& 707 708 709	819 825 831 837 844 880 887 893 899 905 942 948 954 960 967 85 003 009 016 022 028 065 071 077 083 089	850 856 862 868 874 911 917 924 930 936 973 979 985 991 997 034 040 046 052 058 095 101 107 114 120
710 711 712 713 7H	126 132 138 144 150 187 193 199 205 211 248 254 260 266 272 309 315 321 327 333 370 376 382 388 394	156 163 169 175 181 217 224 230 236 242 278 285 291 297 303 339 34$ 352 358 364 400 406 412 418 425
7i5 716 717 718 719	43i 43" 443 449 45? 491 497 503 509 516 552 558 564 570 576 612 618 625 631 637 673 6^ 685 691 697	461 467 473 479 485 522 528 534 540 546 582 588 594 600 606 643 649 655 661 667 703 709 715 721 727
720 721 722 723 724	733 739 745 75* 757 794 800. 806 812 818 854 860 866 872 878 914 920 926 932 938 974 980 986 992 998	763 769 775 781 788 824 830 836 842 848 884 890 896 902 908 944 950 956 962 968 *OO4 #OIO *Ol6 *O22 *O28
72| 726 728 729	86 034 040 046 052 058 094 loo 1 06 112 118 153 159 165 171 177 213 219 225 231 237 273 279 285 291 297	064 070 076 082 088 124 130 136 141 147 183 189 I9j 2OI 2O7 243 249 255 26l 267 303 308 314 320 326
730 73i 732 733 734	332 338 344 350 356 392 398 404^ 410 415 451 457 463 469 475 510 516 522 528 534 570 576 581 587 593	362 368 374 380 386 421 427 433 439 445 481 487 493 499 504 540 546 552 558 564 599 6oj 611 617 623
735 736 737 738 739	629 635 641 646 652 688 694 700 705 711 747 753 759 764 77° 806 812 817 823 829 864 870 876 882 888	658 664 670 676 682 717 723 729 735 741 776 782 788 794 800 835 841 847 853 859 894 900 906 911 917
740 74i 742 743 744	923 929 935 941 947 982 988 994 999 *oo$ 87 040 046 052 058 064 099 105 III Il6 122 157 163 169 175 181	953 958 964 970 976 *on *oi7 #023 #029 ^035 070 07=; 08 1 087 093 128 134 140 146 151 186 192 198 204 210
745 746 747 748 749	' 2l6 221 227 233 239 274 280 286 291 297 332 338 344 349 355 390 396 402 408 413 448 454 460 466 471	245 251 256 262 268 3°3 309 3J3 320 326 361 367 373 379 384 419 425 431 437 442 477 483 489 493 500
750	506 512 518 523 529	535 54i 547 552 558
N.	L. 0 1 2 3 4	56789	P.P.

LOGARITHMIC TABLES

31

N.	L. 0 1 2 3 4	56789	P.P.
750 75i 752 753 754	87506 512 518 523 529 564 57° 576 581 587 622 628 633 639 645 679 685 691 697 703 737 743 749 754 760	535 54i 547 552 558 593 599 604 610 616 651 656 662 668 674 708 714 720 726 731 766 772 777 783 789	6 i 0,6 2 1,2 3 1,8 4 2,4 I 3'2 6 3,6 7 4,2 8 4,8 9 5/4 5 i o/5 2 1,0 3 i/S 4 2,0 5 2,5 6 3,o 7 3,5 8 4,0 9 4/5
757 758 759	793 800 806 812 818 852 858 864 869 875 910 915 921 927 933 967 973 978 984 990 88 024 030 036 041 047	823 829 833 841 846 881 887 892 898 904 938 944 950 955 961 996 *ooi *007 *oi3 *oi8 053 058 064 070 076
760 761 702 763 764	08 i 087 093 098 104 138 144 150 156 161 195 201 207 213 218 252 258 264 270 275 309 3*5 321 326 332	no 116 121 127 133 167 173 178 184 190 224 230 235 241 247 281 287 292 298 304 338 343 349 355 360
7S 766 769	366 372 377 383 389 423 429 434 440 446 480 485 491 497 502 536 542 547 553 559 593 598 604 610 615	393 400 406 412 417 451 457 463 468 474 508 513 519 523 530 564 570 576 581 587 621 627 632 638 643
770 771 772 773 774	649 655 660 666 672 705 711 717 722 728 762 767 773 779 784 818 824 829 833 840 874 880 885 891 897	677 683 689 694 700 734 739 745 75° 75$ 790 795 801 807 812 846 852 857 863 868 902 908 913 919 923
775 776 777 778 779	930 936 941 947 953 986 992 997 *oo3 *oog 89 042 048 053 059 064 098 104 109 113 120 154 159 163 170 176	958 964 969 975 98i *oi4 *02o *025 ^031 ^037 070 076 08 i 087 092 I2O 131 137 143 148 l82 187 193 198 2O4
780 781 782 783 784	2O9 215 221 226 232 265 271 276 282 287 321 326 332 337 343 376 382 387 393 398 432 437 443 448 454	237 243 248 254 26O 293 298 304 310 315 348 354 360 365 371 404 409 413 421 426 459 463 470 476 481
785 786 787 788 789	487 492 498 504 509. 542 548 553 559 564 597 603 609 614 620 653 658 664 669 673 708 713 719 724 730	513 520 526 531 537 57° 575 581 586 592 625 631 636 642 647 680 686 691 697 702 735 741 746 752 757
790 791 792 793 794	763 768 774 779 783 818 823 829 834 840 873 878 883 889 894 927 933 938 944 949 982 988 993 998 *004	790 796 801 807 812 845 851 856 862 867 900 905 911 916 922 953 960 966 971 977 *009 *OI3 *020 *026 *03 I
795 796 797 798 799	90037 042 048 053 059 091 097 102 108 113 146 151 157 162 168 200 206 211 217 222 253 260 266 271 276	064 069 073 080 086 119 124 129 233 140 173 179 184 189 193 227 233 238 244 249 282 287 293 298 304
800	309 314 320 325 331	336 342 347 352 358
N.	L. 0 1 2 3 4	56789	P.P.

32

No. 53— USE OF LOGARITHMS

N.	L. 0 1 2 3 4	56789	P.P.
800 801 802 803 804	90309 3*4 320 32$ 331 363 369 374 380 385 417 423 428 434 439 472 477 482 488 493 526 531 536 542 547	336 342 347 352 358 390 396 401 407 412 445 450 455 461 466 499 504 509 515 520 553 558 563 569 574	6 i 0,6 2 1,2 3 1,8 4 2,4 I 3'2 6 3,6 7 4/2 8 4,8 915,4 5 i 0,5 2 I,O 3 i,5 4 2,0 5 2,5 6 3,o 7 3,5 8 4,0 9 4,5
805 806 807 808 809	580 585 590 596 601 634 639 644 650 655 687 693 698 703 709 741 747 752 757 763 795 800 806 811 816	607 612 617 623 628 660 666 671 677 682 714 720 725 730 736 768 773 779 784 789 822 827 832 838 843
810 811 812 813 814	849 854 859 865 870 902 907 913 918 924 956 961 966 972 977 91 009 014 020 02§ 030 062 068 073 078 084	875 88 I 886 891 897 929 934 940 945 950 982 988 993 998 *004 036 041 046 052 057 089 094 100 105 no
8iS 816 817 818 819	116 121 126 132 137 169 174 180 185 190 222 228 233 238 243 275 28l 286 291 297 328 334 339 344 35o	142 148 153 158 164 196 201 206 212 217 249 254 259 265 270 302 307 3I2 3l8 323 355 36o 36$ 37i 376
820 821 822 g	381 387 392 397 403 434 440 445 450 455 487 492 498 503 508 540 54$ 55i 556 56i 593 598 603 609 614	408 413 418 424 429 461 466 471 477 482 514 519 524 529 535 566 572 577 582 587 619 624 630 635 640
£ 827 828 829 830 831 832 833 834	64$ 651 656 661 666 698 703 709 714 719 751 756 761 766 772 803 808 814 819 824 855 861 866 871 876	672 677 682 687 693 724 730 735 740 745 777 782 787 793 798 829* 834 840 845 850 882 887 892 897 903
908 913 918 924 929 960 965 971 976 981 92 012 018 023 028 033 065 070 075 080 085 117 122 127 132 137	934 939 944 95o 955 986 991 997 *oo2 *007 038 044 049 054 059 091 096 101 106 in 143 148 153 158 163
III 837 838 839	169 174 179 184 189 221 226 231 236 241 273 278 283 288 293 324 330 335 340 345 376 381 387 392 397	195 200 205 210 215 247 252 257 262 267 298 304 309 314 319 35o 35$ 361 366 371 402 407 412 418 423
840 841 842 843 844	428 433 438 443 449 480 485 490 495 500 53i 53° 542 547 552 583 588 593 598 603 634 639 645 650 655	454 459 464 469 474 505 511 516 521 526 557 562 567 572 578 609 614 619 624 629 660 665 670 675 681
845 846 847 848 849	686 691 696 701 706 737 742 747 752 758 788 793 799 804 809 840 845 850 855 860 891 896 901 906 911	711 716 722 727 732 763 768 773 778 783 814 819 824 829 834 865 870 875 88 i 886 916 921 927 932 937
850	942 947 952 957 962	967 973 978 983 988
N.	L. 0 1 2 3 4	56789	P.P.

LOGARITHMIC TABLES

33

N.	L. 0 1 2 3 4	56789	P.P.
850 85i 852 853 854	92942 947 952 957 962 993 998 *oo3 *oo8 #013 93044 049 054 059 064 095 loo 105 no 115 146 151 156 161 166	967 973 978 983 988 *oi8 *024 *029 *034 *039 069 075 080 085 090 120 125 131 136 141 171 176 181 186 192	i 2 3 4 9 I 2 3 4 7 8 9 I 2 3 4 7 8 9	6 0,6 I/2 1,8 2,4 3,<? 3,6 4,2 4,8 5,4 5 o,5 1,0 i,5 2,0 2,5 3,0 3,5 4,o 4,5 4 4 o,4 0,8 1,2 1,6 2,0 2,4 2,8 11
SP 857 858 859	197 202 207 212 217 247 252 258 263 268 298 303 308 313 318 349 354 359 3^4 3^9 399 404 409 414 420	222 227 232 237 242 273 278 283 288 293 323 328 334 339 344 374 379 384 389 394 425 430 435 440 445
860 861 862 863 864	45o 455 460 465 470 50° 5°5 5io 5*5 520 551 556 561 566 571 601 606 611 616 621 651 656 661 666 671	475 480 485 490 495 526 53i 536 54i 546 576 581 586 591 596 626 631 636 641 646 676 682 687 692 697
865 866 867 868 869	702 707 712 717 722 752 757 762 767 772 802 807 812 817 822 852 857 862 867 872 902 907 912 917 922	727 732 737 742 747 777 782 787 792 797 827 832 837 842 847 877 882 887 892 897 927 932 937 942 947
870 871 872 873 874	952 957 962 967 972 94 002 007 012 017 022 052 057 062 OOJ 072 ioi 106 in 116 121 151 156 161 166 171	977 982 987 992 997 027 032 037 042 047 077 082 086 091 096 126 131 136 141 146 176 181 186 191 196
875 876 877 878 879	201 206 211 2l6 221 250 255 260 265 270 300 305 310 315 320 349 354 359 3^4 3^ 399 404 409 414 419	226 231 236 240 245 275 280 285 290 295 325 330 335 340 345 374 379 384 389 394 424 429 433 438 443
880 881 882 883 884	448 453 458 463 468 498 5°3 507 512 517 547 \ 552 557 562 567 596 601 606 611 616 645 650 655 660 665	473 478 483 488 493 522 527 532 537 542 57i 576 581 586 59i 621 626 630 635 640 670 675 680 685 689
885 886 887 888 889	694 699 704 709 714 743 748 753 758 763 792 797 802 807 812 841 846 851 856 861 890 895 900 905 910	719 724 729 734 738 768 773 778 783 787 817 822 827 832 836 866 871 876 880 885 915 919 924 929 934
890 891 892 893 894	939 944 949 954 959 988 993 998 *oo2 *oo7 95 036 041 046 051 056 085 090 095 loo 105 134 139 143 148 153	963 968 973 978 983 *OI2 *OI7 *022 #027 ^032 061 066 071 075 080 109 114 119 124 129 158 163 168 173 177
895 896 897 898 899	182 187 192 197 202 231 236 240 245 250 279 284 289 294 299 328 332 337 342 347 376 381 386 390 395	207 211 2l6 221 226 255 260 265 270 274 303 308 313 318 323 SS2 357 361 366 37i 400 405 410 415 419
900	424 429 434 439 444	448 453 45s 463 468
N.	L. 0 1 2 3 4	56789	P.P.

34

No. 53— USE OF LOGARITHMS

N.	L. 0 1 2 3 4	56789	P.P.
900 901 902 903 904	95424 429 434 439 444 472 477 482 487 492 521 525 530 53$ 540 569 574 578 583 588 617 622 626 631 636	448 453 458 463 468 497 501 506 511 516 545 550 554 559 564 593 598 602 607 612 641 646 650 65$ 660	5 i 0,5 2 1,0 3 i/5 4 2,0 5:2,5 6; 3,0 7 3,5 8 4,0 9:4/5 4 ilo,4 2 0,8 3 1/2 4 1,6 5 2/o 6 2,4 7 2,8 83,2 9 3/6
907 908 909	665 670 674 679 684 713 718 722 727 732 761 766 770 775 780 809 813 818 823 828 856 861 866 871 875	689 694 698 703 708 737 742 746 75i 756 785 789 794 799 804 832 837 842 847 852 880 885 890 893 899
910 911 912 913 914	904 909 914 918 923 952 957 961 966 971 999 *oo4 ^009 #014 *oi9 96047 052 057 061 066 095 099 104 109 114	928 933 938 942 947 976 980 98$ 990 995 #023 #028 #033 $038 #042 071 076 080 085 090 118 123 128 133 137
9i5 916 917 918 919	142 147 152 156 161 190 194 199 204 209 237 242 246 251 256 284 289 294 298 303 332 336 34i 346 35°	166 171 17$ 180 185" 213 218 223 227 232 261 26$ 270 273 280 308 313 317 322 327 35S 360 363 369 374
920 921 922 923 924	379 384 388 393 398 426 431 43$ 440 445 473 478 483 487 492 520 525 530 534 539 567 572 577 581 586	402 407 412 417 421 430 454 459 464 468 497 501 506 511 515 544 548 553 558 562 591 59$ 600 603 609
92J 926 927 928 929	614 619 624 628 633 661 666 670 675 680 708 713 717 722 727 755 759 764 769 774 802 806 811 816 820	638 642 647 652 656 683 689 694 699 703 731 736 741 745 750 778 783 788 792 797 823 830 834 839 844
930 93i 932 933 934	848 853 858 862 867 895 900 904 909 914 942 946 951 956 960 988 993 997 *002 ^007 97035 039 044 049 053	872 876 881 886 890 918 923 928 932 937 963 970 974 979 984 #ou *oi6 *02i *02$ #030 058 063 067 072 077
935 936 937 938 939	08 1 086 090 095 too 128 132 137 142 146 174 179 183 i 88 192 220 225 230 234 239 267 271 276 280 285	104 109 114 118 123 151 155 160 163 169 197 202 206 211 2l6 243 248 253 257 262 290 294 299 304 308
940 941 942 943 944	313 317 322 327 331 359 364 368 373 377 4o§ 410 414 419 424 451 456 460 465 470 497 5°2 506 511 5l6	336 340 345 35o 354 382 387 391 396 400 428 433 437 442 447 474 479 483 488 493 520 523 529 534 539
$ S3 949	543 548 552 557 562 589 594 598 603 607 633 640 644 649 653 68 1 685 690 695 699 727 73i 736 740 745	566 571 575 58o 583 612 617 621 626 630 658 663 667 672 676 704 708 713 717 722 749 754 759 7^3 7^8
950	772 777 782 786 791	795 800 804 809 813
N.	L. 0 1 2 3 4	56789	P.P.

LOGARITHMIC TABLES

35

N.	L. 0 1 2 3 4	56789	P.P.
950 95i 952 953 954	97772 777 782 786 791 818 823 827 832 836 864 868 873 877 882 909 914 918 923 928 955 959 964 968 973	795 800 804 809 813 841 845 850 855 859 886 891 896 900 905 932 937 94i 946 950 978 982 987 991 996
955 956 957 958 959	98 ooo 005 009 014 019 046 050 055 059 064 091 096 100 105 109 137 141 146 150 155 182 186 191 195 200	023 028 032 037 041 068 073 078 082 087 114 118 123 127 132 159 164 168 173 177 204 209 214 218 223
960 961 962 963 964	227 232 236 241 24^ 272 277 281 286 290 318 322 327 331 336 363 367 372 376 381 408 412 417 421 426	250 254 259 263 268 295 299 304 308 313 340 345 349 354 358 385 390 394 399 4°3 430 «5 439 444 448	5 ijo,5 2 1,0 3 i,S
965 966 967 968 969	453 457 462 466 471 498 502 507 511 516 543 547 552 556 561 588 592 597 601 605 632 637 641 646 650	475 480 484 489 493 520 525 529 534 538 565 57° 574 579 583 610 614 619 623 628 655 659 664 668 673	4 2,0 5 2,5 6 3/o 7 3,5 8 4,0 9 4,5
970 971 972 973 974	677 682 686 691 695 722 726 731 735 740 767 771 776 780 784 8il 816 820 825 829 856 860 865 869 874	700 704 709 713 717 744 749 753 758 762 789 793 798 802 807 834 838 843 847 851 878 883 887 892 896
975 976 977 978 979 980 981 982 983 984	900 905 909 914 918 945 949 954 958 963 989 994 998 #003 *oo7 99034 038 043 047 052 078 083 087 092 096 123 127 131 136 140 167 171 176 180 185 211 2l6 22O 224 229 255 260 264 269 273 300 304 308 313 317	923 927 932 936 941 967 972 976 981 985 #012 *oi6 #021 #02=; #029 056 061 065 069 074 100 105 109 114 118 145 149 154 158 162 189 193 198 202 207 233 238 242 247 251 277 282 286 291 295 322 326 330 335 339	4 I 0,4 2 0,8 3 1,2
985 986 987 988 989	344 348 352 357 361 388 392 396 401 4°5 432 436 44i 445 449 476 480 484 489 493 520 524 528 533 537	366 370 374 379 383 410 414 419 423 427 454 458 463 467 471 498 502 506 511 515 542 546 550 555 559	4 1,6 5 2,0 6 2,4 7 2,8 8 3,2 9 3,6
990 991 992 993 994	564 568 572 577 581 607 612 616 621 625 651 656 660 664 669 695 699 704 708 712 739 743 747 752 756	585 590 594 599 603 629 634 638 642 647 673 677 682 686 691 717 721 726 730 734 760 765 769 774 778
995 996 997 998 999	782 787 791 79$ 800 826 830 835 839 843 870 874 878 883 887 913 917 922 926 930 957 961 965 97° 974	804 808 813 817 822 848 852 856 861 86^ 891 896 900 904 909 935 939 944 948 952 978 983 987 99i 996
1000	ooooo 004 009 013 017	022 026 030 035 039
N.	L. 0 1 2 3 4	56789	P.P.

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LD

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LD 21-100m-9,'47(A5702sl6)4

Bf 18

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CONTENTS OF DATA JHEI-T BOOKS

. i. Sc-ew Threads.— T nired

ail- '

>«i; C

Bri£- gs Pipe 'i • G; ^es: Fire Hose T -read: AYorm orms . Machine. "vVuod, C^:-r:iare Brit

<M>IS; Tape] Tnrr.i g: Chang* ing for <'.;e Latne, Borin:, B: -s and

No. 11. Milling Marine Indexin •, damping Devices and. Plasar Jacks. —

r- Mes for Milling Ala-nine Indexing, Ci. .use Gears foi 'nilhig- Spirals; foi it t ting I.'.-Jexi.ig P"<v4d when to

	Jolts antl NntJ. — Fil-	Clutches, Jig '-urni 1^0 vices; .-
	-he;n	and Clamp?; I'lai^t ' ^ :KS.
	x w, • St.°n<l- .11	No. lv' fcT'j j.i'- * Pipcj Pittings. ads a . lag " -iron Fii .ze Fii v "s;
		jJen<J|, Pi'>e .-imp
	• I .Scr---\vh ; TJ- .) Drills;	:is of Pipe, foi Vc . .Services,
	.	io. 13. Boilers *af 'aimncys.
»0	. ~ 0.- P. "i- 3. — B • i, iVlachi; ,	rfpac'ig iul Bracing i 'lers: St
	-	of Boiler Joints; Rive'. ^ Boiler S>
L«i(	,.0-b- .- .v Mr ne	Chimneys
Tat <	r il jr "; s;	No. 14. I-ocon otive inf Railway "a.i.
Su.j	t-oU ' ns;	--Lr ITT i "ive B"il •!•<?; ^ aring Pr ,..•.
Pi	Hi -i " , 'itu •	>r L,^ ?• moth )t^ a
A<-	• • j«k '! '< ding	L.la« -ificei; ins. Rnij Sections:
:_i	^^^^H	Swit-, 'iep am; " iv. -. T
	IT ;t .v Dr^il and	Force; i.iertin • ? ' ins. Brake 1
,1	T Faell	Br. ';. -. Rods, etc.
	ij Pi- .	o. 15. Steam a ^ das _ ng-ines uraied Stej.ni; • :• m Pipe Si >s;
	Ja • .. >1 i:	Engine .f Cyl -r
	r-	StuiTiin.u .Setting Corliss . .
*'	vf	Vaj Air i
s	;: !.	l)cl 1 •El!.,,
	C-arir^r. — Piameir-.	Automobile >"' ^in • •'|:.v'i,ats,
		No. 16. MatV.m^ ,i.-. hies. S of Mixe<: .\";
	-1 Cast-iron C.-	Circles; 'fahV: Solution of Tr. • ;? . Regu, P ". j jiK-
	; el. fc> *' ' -. d 'Vorm Q-*iai'-	gression, cio.
t?	and " j for Bavel •! <.-ars; \.-	»o. li >*fichanics a^-I Strength J terials.-fW- k: K re>; Cent
		Force:
		:
	Spiral	.Materials: • ' i
	lj.0 I'll1	i io of t /lit!-"'1 :> , 1 -0
		lers
"!fC	Cey \ie Keyvrays. —	Jfo. 18. y?e; F <••'• v... ? r'ci .al
	» t Til). Mti— "liis R,71u	Desi^ti.— • 1-j
	tuiafrtit ' *e 1	uli 01 1
	ni Runnn ,	Ni
		'• v: «p!'' o ti.ii ! I
	lilling K»-y-	•
		1* . .9. Belt, B ne
	^*< aring* Couplings, ClntclieK,	D'i., nifiioi.
	.haiu and ^iooks. — Pil:	le\
	i
	(.".amp t ou: ing	ti'..;isin: Dr;
Ci:.	"uni Scor«. .-.	an.
V.' '	.ngs, TMdes r.ud Machine	No. 20. Wiri i .grains. «eat3 r - I Ventilation, n xscellaaeous Ti les 1
		Typl<
11	-	I! OC( Ci. I
		'•'a 1
		on.-- !-^; Cent 'f;:g I
*	10 . ^.o.f * .ive. Speeds and Peeds,	lot -,'v <• I
v»hi	i a crearint a -in^ Uars.
	at ling	j-si '•
		Ta- les, Wei
		• ftii, -i.>oil
	mechanical	jf.'irnal, o;. gins 'or u< tbc R-ferenc< anl
	.. published in foui	litioTi •- hp /S/it •< Edition. ^>L.OO a. .'a;B
	oO a year;	fbe '\00. a year, u- trl
;	-	fl
^^•fcad'j , trial Press, Publishers o^ ' -"BINARY,
-g.		; ex York City, TJ S. 1